230 Proceedings of Philosophical Societies. [Makch^ 



axis is more or less turned, even when the position of the crystal 

 which serves to analyse the light is unchanged, we see the colours 

 of the rays which it polarizes succeed each other apparently 

 without any law. But all these anomalies are only apparent ; 

 they assume the character of the most perfect regularity when 

 we observe them with method, and measure them exactly. 



But we must here, as we did under the perpendicular inci- 

 dence, distinguish carefully the intensity and the colour of the 

 pencil which the plates polarize. The intensity follows a law 

 independent of the colour^ and the colours a law independent of 

 changes in the intensity. 



The fundamental law of the intensities is the following. If ■ 

 we set out from any position of the plate in which the intensity 

 of the pencil which it polarizes is o, or at least in which it is 

 confounded with the primitive polarization, and if without 

 changino the inclination of that plate, we make it turn round 

 the polaiize dray , so that the plane of incidence of the ray upon 

 its surface describes thus an angle a, comprehended between o and 

 90°, the pencil polarized hy the plate ivill re-appear, and sepa- 

 rate from the rest of the transmitted light, in the direction of its 

 polarization ; hut it will disappear again if without changing 

 the inclination or the azimuth of the plane of incidence round the 

 polarized ray, we turn the axis of the plate in its plane in such _ 

 a way as to describe upon the plane an angle a equal and con- 

 trary to that which the plane of incidence described. 



This perfect compensation of the two angles thus measured in 

 different planes, is a very singular phenomenon, which we shall 

 find to result from the theory of Biot. 



We shall now state the formulas to which this law corresponds. 

 Let A be the dihedral angle which the primitive plane of 

 polarization forms with the plane of incidence of the ray upon 

 the plate. Let i denote the angle which the axis of the plate 

 forms upon its surface with the trace of the plane of incidence, 

 this angle being reckoned in an opposite direction from the 

 preceding. Let O, as formerly, be the intensity of the pencil 

 which preserves its primitive polarization in traversing the plate, 

 E the intensity of the pencil to which it gives a new polarization. 

 Finally, let a be the dihedral angle which the principal section 

 of the rhomboid which serves to analyse the light forms with the 

 primitive plane of polarization. 



If we name F^, F^, the two ordinary and extraordinary 



pencils given by the rhomboid, we shall have 



F^ = O cos.2 a + E cos.^ [2 (i - A) - ^J. 



F^ = O sin.? a + E sin.2 [2 (i - A) - «]. 



When the incidence is perpendicular i A becomes the 

 right azimuth of the axis of the plate in relation to the primitive 



flane of polarization, and we get those formulas which havr^ 

 een former])^ stated, 



